Integration
Formulas List can be categorized into General Integral Formulas, Rational
and Irrational Functions Integrals, Exponential & Logarithmic Functions
Integrals and Trigonometric Functions Integral formulas. Table
of Integration Formulas is as follows:
General Integral
Formulas
Indefinite Integrals: Substitution
method, Integral [f [g(x)] g’(x)] dx = Integral [f (u)] du
Indefinite Integrals:
Integration by parts, Integral [f(x) g’(x)] dx = f(x) g(x) – integral [g(x) f’(x)]
dx
Rational and
Irrational Functions Integral Formulas
Integral[c] dx = cx +c (c
is some constant)
Integral [x^n] dx = x^ (n+1)/ (n+1)
Integral [1/x] dx = ln|x| + c
Integral [sqrt(x)] dx = 2x [sqrt(x)]/3 + c
Integral [1/ (1+x^2)] dx = arc tan(x) + c
Integral [1/sqrt (1-x^2)] dx = arc sin(x) + c
Exponential and
Logarithmic Functions Integral Formulas
Integral [ln x] dx = x ln x – x + c
Integral [1/ln x] dx = ln|ln x| + ln x + summation (n=2 to
infinity) {[ln x] ^i/i.i!}
Integral [a^x] dx =a^x/ [ln a] + c
Integral [e^x] dx= e^x + c
Integral[x^n. e^ (cx)] dx = (1/c) e^ (cx) – (n/c) Integral[x^
(n-1). e^ (cx)] dx
Integral[x^n ln x] dx =[x^ (n+1)]/ [n+1]. In x – [x^ (n+1)/
(n+1) ^2] + c
Trigonometric
Integral Formulas
Integral [ sin x] dx = - cos x + c
Integral[cos x] dx = sin x + c
Integral[tan x] dx = ln|sec x| + c
Integral[csc x]dx = - ln|csc x + cot x| + c
Integral[sec x] dx = ln |tan x + sec x| + c
Integral[cot x]dx = ln|sin x| + c
Integral[sin^2 (x)] dx = ½[x – sin x cos x] + c
Integral[cos^2(x)]dx = ½[x + sin x cos x] + c
Integral[tan^2(x) dx = tan x – x + c
Integral[sec^2(x)]dx = tan x + c
Integral[csc^2(x)]dx = - cot x + c
Integral[sin x. cos x]dx = -(1/4) cos 2x
Integral[sin^n(x). cos x] dx = - [cos^(n+1)x/(n+1)]
Integral[sin x. cos^n(x)]dx = [sin^(n+1) x/(n+1)]
Integral[arc sin(x)]dx= x arc sin(x) + [sqrt(1-x^2)] + c
Integral[arc csc(x)]dx = x arc cos(x) – [sqrt(1-x^2)]+c
Integral[arc tan(x)]dx = x arc tan(x) – (1/2) ln(1+x^2) + c
Integral[sinh(x)]dx = cosh(x) + c
Integral[cosh(x)]dx = sinh(x) + c
Reduction Formulas
Reduction Formulas are the Antiderivatives. Following are
the various Reduction Formulas
Integral[ln(x)]^n dx = x [ln (x)]^n - Integral[ln(x)^(n-1)]dx
Integral[x^(n). e^(ax)]dx = (1/a)x^(n). e^(ax) – (n/a)
Integral[x^(n-1). e^(ax)]dx
Integral[sin^n(x)]dx =
- (1/n)[sin^(n-1) x. cos(x)] + [(n+1)/n]Integral[sin^(n-2) x] dx
Integral[tan^n(x)]dx = [tan^(n-1) x/(n-1)] –
Integral[tan^(n-2) x ] dx
Integral[sec^n(x)]dx = [sec^(n-2) x.tan x/(n-1)] +
[(n-2)/n-1)]Integral[sec^(n-2) x] dx
Integral[x^n. sin(x)]dx = - x^n. cos(x) + n
Integral[x^(n-1). cos(x)]dx
Integarl[x^n. cos(x)]dx = x^n. sin(x) – n Integral[x^(n-1).
Sin(x)] dx
Integral[sin^n(x). cos^m(x)]dx
=
[sin^(n+1) x. cos^(m+1) x]/n+m +
[m-1/n+m]Integral[sin^n(x) cos^(m-2)x] dx
Integral [sin^n(x). cos^m(x)]dx
=
[sin^(n-1) x. cos^(m+1) x]/n+m +
[n-1/n+m]Integral[sin^(n-2)x. cos^(m)x] dx
Integral[sec^n(x)]dx = [1/(n-1)] sec^(n-2)x. tan
x + [(n-2)/(n-1)][sec^(n-2)x] dx